Download Sample Exam 1 - Purdue Engineering

ECE 302: Probabilistic Methods in Electrical and Computer Engineering
Fall 2015
Instructor: Prof. A. R. Reibman
Sample Exam 1
Fall 2015, MWF 2:30-3:20pm
(SAMPLE)
This is a closed book exam. Neither calculators nor help sheets are allowed.
Cheating will result in a zero on the exam and possibly failure of the class. Do not cheat!
Put your name on every page of the exam and turn in everything when time is up.
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I certify that I have neither given nor received unauthorized aid on this exam.
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Problem 1. (Multiple choice: 5 points)
Bill is a physicist with probability 0.6, an amateur jazz musician with probability 0.7, and is neither
with probability 0.25. Determine the probability that he is a physicist or an amateur jazz musician
but not both.
(a) 0.1
(b) 0.2
(c) 0.3
(d) 0.5
(e) Impossible to determine based on the information given.
(f) None of the above.
Problem 2. (Multiple choice: 5 points)
You roll two fair dice, and record the sum of the number of dots facing upward. What is the
probability the sum of the two is even and at least one of the dice shows 5.
(a) 1/2
(b) 11/36
(c) 2/3
(d) 5/36
(e) Impossible to determine based on the information given.
(f) None of the above.
Problem 3. (Always/Sometimes/Never (5 points each, 20 points total))
Events E, F, and G form a list of mutually exclusive and collectively exhaustive events with P (E) 6=
0, P (F ) 6= 0, and P (G) 6= 0. Determine for each of the following statements, whether it must be
true (always), it might be true (sometimes), or it cannot be true (never). Mark your answer clearly
to the left of each question.
(a) E c , F c and Gc are mutually exclusive.
(b) E c , F c and Gc are collectively exhaustive.
(c) P (E c ) + P (F c ) > 1
(d) P (E c ∪ (E ∩ F c ) ∪ (E ∩ F ∩ Gc )) = 1
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Problem 4. (5 points each, 20 points total)
Consider the x-y plane. Consider a disk (circle) of radius 2 on the plane, which are those points
satisfying x2 + y 2 ≤ 2. We throw a dart at the disk and we know that the dart will land uniformly
likely on the disk. Let X and Y denote the x and y coordinates of the landing location of the dart.
Answer the following questions:
(a) Consider an event A = {X 2 + Y 2 ≥ 1}. What is the probability of the event A?
(b) Consider an event B = {X + Y > 0}. What is the probability of the event B?
(c) Are events A and B independent?
(d) Consider an event C = {X > 1}. Are events A and C independent?
Problem 5. (10 points)
Two fair coins are each flipped once. A trustable friend tells you “I saw at least one of them was
heads”. Use conditional probabilities to determine the probability that both coins came up heads,
given that at least one was heads.
Problem 6. (5 points each, total 10 points)
Throw a fair dice and toss a fair coin. Let X and Y denote the outcomes of the dice and the coin
respectively, where we use the convention that Y = 1 if the outcome of the coin is head. Y = 0 if
the outcome of the coin is tail.
(a) What is the sample space in this experiment?
(b) What is the probability that X 2 + Y is a prime number? (Note that 1 is NOT a prime
number. The smallest prime number is 2.)
Problem 7. (10 points each, 30 points total)
Zeros and ones are sent over a noisy communication channel, where the transmission of each bit can
be considered to be independent sequential experiements. The probability that each 0 is correctly
sent is 0.9, while the probability that each 1 is correctly sent is 0.85. The digit 0 is sent with
probability 0.6.
(a) Find the probability that an error occurs, for each bit sent.
(b) Given that you detect a 1, what is the probability that a 1 had been sent.
(c) If the string 0010 is sent, what is the probability the string is correctly received.
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